Double metric, generalized metric, and α′-deformed double field theory

Abstract

We relate the unconstrained “double metric” of the “α′-geometry” formulation of double field theory to the constrained generalized metric encoding the spacetime metric and b-field. This is achieved by integrating out auxiliary field components of the double metric in an iterative procedure that induces an infinite number of higher-derivative corrections. As an application, we prove that, to first order in α′ and to all orders in fields, the deformed gauge transformations are Green-Schwarz–deformed diffeomorphisms. We also prove that to first order in α′ the spacetime action encodes precisely the Green-Schwarz deformation with Chern-Simons forms based on the torsionless gravitational connection. This seems to be in tension with suggestions in the literature that T-duality requires a torsionful connection, but we explain that these assertions are ambiguous since actions that use different connections are related by field redefinitions.